Solid harmonics

In mathematics, solid harmonics are defined as solutions of the Laplace equation in spherical polar coordinates. There are two kinds of solid harmonic functions: the regular solid harmonics, which vanish at the origin, and the irregular solid harmonics, which have an singularity at the origin. Both sets of functions play an important role in potential theory. Regular solid harmonics appear in chemistry in the form of s, p, d, etc. atomic orbitals and in physics as multipoles. Irregular harmonics appear in the expansion of scalar fields in terms of multipoles.

It can be shown by expression of L in spherical polar coordinates that L² does not contain a derivative with respect to r. Hence upon division of L² by r² the position of 1/r² in the resulting expression is irrelevant.
After these preliminaries we find that the Laplace equation ∇² Φ = 0 can be written as

Substitution of Φ(r) = F(r) Yml into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution,

The particular solutions of the total Laplace equation are regular solid harmonics:

and irregular solid harmonics:

Racah's normalization (also known as Schmidt's semi-normalization) is applied to both functions

(and analogously for the irregular solid harmonic) instead of normalization to unity. This is convenient because in many applications the Racah normalization factor appears unchanged throughout the derivations.

A more interesting relationship follows from the observation that the regular solid harmonics are homogeneous polynomials in the components x, y, and z of r. We can replace these components by the corresponding components of the gradient operator ∇. Thus, the left hand side in the following equation is well-defined:

By a simple linear combination of solid harmonics of ±m these functions are transformed into real functions. The real regular solid harmonics, expressed in Cartesian coordinates, are homogeneous polynomials of order l in x, y, z. The explicit form of these polynomials is of some importance. They appear, for example, in the form of spherical atomic orbitals and real multipole moments. The explicit Cartesian expression of the real regular harmonics will now be derived.